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Bohm Trajectories for the Noncentral Hartmann Potential
Bohm Trajectories for the Noncentral Hartmann Potential
A noncentral potential in spherical polar coordinates depends on the angular variables and [1, 2]. The Hartmann potential , depending on the variables and , was introduced in quantum chemistry [1] to describe ring-shaped molecules, such as benzene. It can also be applied to the interaction between distorted nuclei. The Hartmann potential adds to the three-dimensional Kepler–Coulomb potential θ term, which corresponds to coupling with the radial degree of freedom [3] with a coupling constant .
θ
ϕ
V(r,θ)
r
θ
κ
2
r
0
V
0
2
r
2
sin
κ
The minimum of the potential well , the coupling constant and the radial distance of the potential minimum from the center of the potential ring are three system-specific variables. The motion of a quantum particle under the influence of the Hartmann potential can be solved exactly in closed form, in terms of Laguerre and Jacobi polynomials for radial and angular parts, respectively.
V
0
κ
r
0
In this Demonstration, the de Broglie–Bohm approach for a Hartmann potential in a superposition eigenstate is investigated. The potential is written in the following form, in atomic units:
ψ
V(r,θ)=θ-=θ-
1
2
V
0
κ
2
r
0
2
r
2
sin
2
r
0
r
1
2
β
2
r
2
sin
α
r
For , the potential reduces to the three-dimensional Kepler–Coulomb potential of the hydrogen atom.
κ=0
In de Broglie–Bohm theory, the particle has a well-defined trajectory in configuration space, calculated from the gradient of the phase function. An appealing feature of the Bohmian description is that one can impose initial positions for the trajectories, just as for a classical dynamical system, in which the final positions of the particles are determined by their initial positions. Such initial positions are not controllable by the experimenter, so there is an appearance of randomness in the pattern of detection. The wavefunction guides the particles in such a way that the particles are attracted to the regions in which the wave density is large [6, 7]. In the regions with small wave density, the particles will be accelerated.
The graphics show the wave density (if enabled), the velocity vector field (red arrows), the initial starting points of eight possible orbits (red points, shown as small red spheres), the actual position (colored points, shown as small spheres) and eight possible trajectories with the initial distance .
δx